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JWBK119-14
212 Data Transformation for Geometrically Distributed Quality Characteristics
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geometric-type control chart or G chart is very useful when the underlying distri-
bution is neither binomial nor Poisson and when a geometric distribution is a better
description of the process behavior.
In particular, when the process is of very high quality, control charts can be devel-
oped for the cumulative count of conforming items between two nonconforming ones.
The cumulative count of conforming items follows a geometric distribution. Calvin 2
first introduced the idea of tracking cumulative counts to monitor a near-zero-defect
or very-high-yield process. The idea was developed into a charting technique and
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named by Goh. Glushkovsky studied the use of the G chart for process monitoring
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of geometrically distributed quantities, and Kaminsky et al. also discussed the use of
the geometric distribution in process monitoring. A number of examples can be found
in these papers. Additional results related to the issue can be found elsewhere. 5−8
The traditional 3σ limits for the G chart, similar to those of the p chart or c chart,
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have been used by Kaminsky et al. Although 3σ control limits are convenient to use,
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it has been recently pointed out by Xie and Goh that they are not appropriate because
the geometric distribution is highly skewed. Furthermore, the normal assumption is
never adequate for approximating a geometric distribution because the probability
density function is always decreasing and hence cannot be bell-shaped like a normal
distribution. Because of the lack of normality, supplementary run rules cannot be
applied directly. Other more advanced monitoring techniques such as the standard
cumulative sum (CUSUM) or exponentially weighted moving average (EWMA) chart
should not be used either. The non-normality problem limits the use of the G chart.
Proper transformation of the raw data needs to be studied.
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A procedure is proposed by Quesenberry and named the geometric Q chart. It is a
form of standardized G chart. By using the transformation, the problem of detecting
changesinthegeometricdistributionistransformedintooneofmonitoringanormally
distributed variable so that other well-developed techniques such as supplementary
run rules, CUSUM and EWMA control schemes can be used. However, the geometric
Q chart control scheme assumes that the parameter is known or already estimated.
Furthermore,theinversenormalisusedandthisisnoteasytoimplementandinterpret
in practice as a plotted point has no direct meaning.
In this chapter, two other possible transformations are studied and compared with
the Q-transformation. One is the simple log transformation which is a limiting case
of the so-called arcsin transformation for non-normal quantities. This transformation,
however, is generally not satisfactory for a geometric distribution. In line with the
square root transformation for positively skewed distributions a double square root
transformation is proposed, because using a single square root transformation the
result is still positively skewed. Numerical results show that the normality after this
transformation is usually better than with other methods. Furthermore, because the
Q-transformation assumes that the model parameter is known or estimated, it is not
very robust, as shown by a sensitivity analysis of the Q-transformation in the last
section of this chapter.

14.2 PROBLEMS OF THREE-SIGMA LIMITS FOR THE G CHART

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The idea of 3σ limits for the G chart is based on the normal approximation. When
the quality characteristic of interest is geometrically distributed with parameter p, the